Golden Resource 5040 Price Interval

Abstract

Every price strictly between the two adjacent layer thresholds makes 5040 the unique maximizer of the golden resource objective.

Theorem 1.1 (The open threshold interval suffices for unique optimality).

$$\forall lambda\in \mathbb{R},n\in \mathbb{N},(\frac{log\left(\frac{12}{11}\right)}{log\left(11\right)}<lambda\wedge \left(lambda<\frac{log\left(\frac{31}{30}\right)}{log\left(2\right)}\wedge 1\le n\right))\Rightarrow (goldenResourceObjective\left(lambda,n\right)\le goldenResourceObjective\left(lambda,5040\right)\wedge \left(goldenResourceObjective\left(lambda,n\right)=goldenResourceObjective\left(lambda,5040\right)\iff n=5040\right))$$

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenResource5040PriceInterval.golden_resource_5040_unique_maximum_of_price_interval (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let lambda lie strictly above log(12/11)/log(11) and strictly below log(31/30)/log(2). For every positive natural number n, the golden resource objective at n is at most its value at 5040, and equality holds exactly when n is 5040.

The proof identifies the upper endpoint with the adopted layer (2,4) and the lower endpoint with the first omitted layer (11,1). Strict decay within each prime and a uniform bound for larger primes propagate these comparisons to every adopted and omitted layer. Strict local threshold maximality is then summed over the union of the prime supports of n and 5040.

This theorem proves only the sufficient open-interval direction. It does not prove necessity outside the interval, classify endpoint ties, supply decimal approximations, compare classical sequences, or interpret the separate price lambda = 0.04 example.

References

• Truth anchor: D5/S3/Arith/GoldenResource5040PriceInterval.golden_resource_5040_unique_maximum_of_price_interval
• Dependency: D5/S3/Arith/GoldenLayerMarginalDecay
• Dependency: D5/S3/Arith/GoldenResourceObjectiveFactorization